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an arithmetical progression has positive terms .the ratio of the difference of the 4th and 8th term to the 15th term is 4/15 and square of the difference of the 4th and the 1st term is 255.which term of the series is 2015?? 
options 
1)225 2)404 3)403 4)410 
please answer

 Aug 24, 2015

Best Answer 

 #1
avatar+128079 
+5

I think that the problem is supposed to say that "square of the difference of the 4th and the 1st term is 225"  rather than 255......when I tried 255 I get a strange square root which doesn't allow for any "integer" answers [If I didn't make a mistake, that is !!!!].....I'm working this one assuming 225 is correct...

The first term is just a1

Let the 4th  term be given by:

a4 = a1 + d(4 -1)   =  a1 + 3d

And the difference between the 4th and 1st terms is just 3d

And the square of this difference  = 225

So  (3d)^2  = 225 →  9d^2 = 225  →  d^2  = 225/9  →   d =  15/3   so d =5 [since all the terms are positive]

 

And we're told that    [ (a1 + d(7) ) - (a1 + d(3)) ] / [a1 + d (14) ]  = 4/15  simplify   

[d(7) - d(3)] /[ a1 + d(14)] = 4/15   and sustituting 5  for d we have

[5(7) - 5(3)] /[ a1 + 5(14)] = 4/15

5*4  = [4/15]  (a1 + (5)*14)     multiply through by 15

5*4*15 = 4[a1 + 70]

300 = 4a1 + 280

20 = 4(a1)

So a1  = 5

 

So

2015 = 5 + 5(n -1)

2010 = 5(n -1)

402 = n - 1    add 1 to both sides

403    which is answer "3"

 

 

 Aug 24, 2015
 #1
avatar+128079 
+5
Best Answer

I think that the problem is supposed to say that "square of the difference of the 4th and the 1st term is 225"  rather than 255......when I tried 255 I get a strange square root which doesn't allow for any "integer" answers [If I didn't make a mistake, that is !!!!].....I'm working this one assuming 225 is correct...

The first term is just a1

Let the 4th  term be given by:

a4 = a1 + d(4 -1)   =  a1 + 3d

And the difference between the 4th and 1st terms is just 3d

And the square of this difference  = 225

So  (3d)^2  = 225 →  9d^2 = 225  →  d^2  = 225/9  →   d =  15/3   so d =5 [since all the terms are positive]

 

And we're told that    [ (a1 + d(7) ) - (a1 + d(3)) ] / [a1 + d (14) ]  = 4/15  simplify   

[d(7) - d(3)] /[ a1 + d(14)] = 4/15   and sustituting 5  for d we have

[5(7) - 5(3)] /[ a1 + 5(14)] = 4/15

5*4  = [4/15]  (a1 + (5)*14)     multiply through by 15

5*4*15 = 4[a1 + 70]

300 = 4a1 + 280

20 = 4(a1)

So a1  = 5

 

So

2015 = 5 + 5(n -1)

2010 = 5(n -1)

402 = n - 1    add 1 to both sides

403    which is answer "3"

 

 

CPhill Aug 24, 2015

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